OFFSET
1,1
COMMENTS
k^3 + 1 is a term iff k + 1 and k^2 - k + 1 are both prime.
Is the sequence infinite? This is an analog of Landau's 4th problem, namely, are there infinitely many primes of the form k^2 + 1?
In other words: are there infinitely many primes p such that p^2 - 3*p + 3 is also prime? - Charles R Greathouse IV, Jul 02 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1400
Eric Weisstein's World of Mathematics, Semiprime
Wikipedia, Semiprime
Wikipedia, Landau's problems
EXAMPLE
9 = 3*3 = 2^3 + 1 is the first semiprime of the form n^3 + 1, so a(1) = 9.
MATHEMATICA
L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1
Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* Zak Seidov, Jun 26 2017 *)
PROG
(PARI) lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", ")); ); \\ Michel Marcus, Jun 27 2017
(PARI) list(lim)=my(v=List(), n, t); forprime(p=3, sqrtnint(lim\1-1, 3)+1, if(isprime(t=p^2-3*p+3), listput(v, t*p))); Vec(v) \\ Charles R Greathouse IV, Jul 02 2017
(Magma) IsSemiprime:= func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // Vincenzo Librandi, Jul 02 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Feb 02 2014
STATUS
approved